What is Linear Algebra?

Wikipedia's entry on linear algebra has a good summary:

Linear algebra is a branch of mathematics concerned with the study of vectors, with families of vectors called vector spaces or linear spaces, and with functions that input one vector and output another, according to certain rules. These functions are called linear maps (or linear transformations or linear operators) and are often represented by matrices. Linear algebra is central to modern mathematics and its applications. An elementary application of linear algebra is to the solution of a system of linear equations in several unknowns. More advanced applications are ubiquitous... . It has extensive applications in engineering, physics, natural sciences and the social sciences. [Linear algebra is also important to study because] nonlinear mathematical models can often be approximated by linear ones.

The Roles of Linear Algebra

There are considerable differences among linear algebra courses and textbooks. Roughly speaking, there are three not mutually exclusive roles that linear algebra plays in the math curriculum:

  1. Linear algebra teaches a set of techniques for solving problems. These classes often emphasize applications.
  2. Linear algebra serves as a bridge toward more advanced or abstract mathematics.
  3. Linear algebra can be an entry point for numerical mathematics. This includes:
    • Using a computer to do mathematics and solve problems.
    • Analyzing the run-time and convergence properties of algorithms, writing original code.

This course will combine elements of 1, 2, and the first item of 3. I'm not certain of the exact balance; this will depend on the interests of the class.

Informal Description

Here is some additional information and advice that should give you a better idea what to expect from this class, how to enjoy it and do well, and help you decide whether or not this class is for you.

  • Linear algebra is standard topic in the college mathematics curricula. It is usually taken by students in their sophomore year. Linear Algebra is required for math, physics, engineering, statistics, and economics majors. I suspect it's required for chemistry majors, but I'm not sure. In general, anyone who works in a mathematical field will need to know linear algebra.

  • My experience as a student was that linear algebra seemed very easy. So I didn't do much work in the class. But then it got harder quite quickly and I was lost and in a lot of trouble. As always, it's important to stay caught up, because you can get left behind pretty quickly in a course like this.

  • Also, when I took linear algebra I remember thinking that the material was kinda silly and that I would never use it. However, I ended up using linear algebra in almost every math and physics class I took, and I use it in my research all the time---much more than I use calculus. Some of the applications of linear algebra might not be apparent immediately, but please believe me when I say that it is a very useful topic to know.

  • This class will move at a fairly brisk pace. I suspect that there will be times when you will need to read a chapter on your own and then ask questions during class if you have any. I think our textbook is pretty good, so learning a few things on your own won't be a problem. (It's also good practice for learning independently.) If we can do this, we'll be able to cover more topics, and I'll also be able to spend time presenting interesting applications of the material.

  • This class does not rely on a knowledge of calculus. Nevertheless, Calc I and II are pre-requisites, as a certain level of comfort with mathematics and abstraction is important background for this course.

  • Introductory linear algebra is at times an odd mixture of tedious and straightforward calculations and austere and potentially difficult abstractions.

  • This is the third time I've taught this class, and this is the first time using this text. So I'm not 100% certain how much we'll be able to cover. Your honest feedback on the pace and level of difficulty will be important.